American Institute of Mathematical Sciences

Advanced
August  2015, 35(8): 3707-3719. doi: 10.3934/dcds.2015.35.3707

Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities

Mitsuru Shibayama 1,

1. 

Division of Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

Received  April 2014 Revised  December 2014 Published  February 2015

It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems. We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systems with two degrees of freedom. The homogeneous degree can be taken from real values (not necessarily integer). The proof is based on the blowing-up theory which McGehee established in the collinear three-body problem. We also compare our result with Molares-Ramis theory which is the strongest theory in this field.
Keywords: blowing-up technique, homogeneous Hamiltonian systems., Integrability.
Mathematics Subject Classification: Primary: 37J30; Secondary: 70F1.
Citation: Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707
References:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Springer, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

H. Bruns, Über die Integrale des Vielkörper-Problems, Acta Math., 11 (1887), 25-96. doi: 10.1007/BF02612319.  Google Scholar

[3]

R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017.  Google Scholar

[4]

R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem, Celestial Mech., 28 (1982), 25-36. doi: 10.1007/BF01230657.  Google Scholar

[5]

G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615. doi: 10.3934/dcds.2014.34.4589.  Google Scholar

[6]

S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. doi: 10.1007/BF02592182.  Google Scholar

[7]

R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175.  Google Scholar

[8]

R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM J. Math. Anal., 15 (1984), 857-876. doi: 10.1137/0515065.  Google Scholar

[9]

J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser Basel, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[10]

J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar

[11]

H. Poincaré, New Methods of Celestial Mechanics Vol. 1, American Institute of Physics, 1993.  Google Scholar

[12]

M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates, J. Phys. A: Math. Gen., 30 (1997), 5869-5876. doi: 10.1088/0305-4470/30/16/026.  Google Scholar

[13]

M. Shibayama, Non-integrability of the collinear three-body problem, Discrete Contin. Dyn. Syst., 30 (2011), 299-312. doi: 10.3934/dcds.2011.30.299.  Google Scholar

[14]

M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Nonlinearity, 22 (2009), 2377-2403. doi: 10.1088/0951-7715/22/10/004.  Google Scholar

[15]

H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems, Physica, 21 (1986), 163-170. doi: 10.1016/0167-2789(86)90087-4.  Google Scholar

[16]

H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9.  Google Scholar

[17]

M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians, RIMS Kokyuroku Bessatsu, B40 (2013), 177-189.  Google Scholar

[18]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I. Funktsional. Anal. i Prilozhen., 16 (1982), 30-41, 96.  Google Scholar

[19]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II. Funktsional. Anal. i Prilozhen., 17 (1983), 8-23.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Springer, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

H. Bruns, Über die Integrale des Vielkörper-Problems, Acta Math., 11 (1887), 25-96. doi: 10.1007/BF02612319.  Google Scholar

[3]

R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017.  Google Scholar

[4]

R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem, Celestial Mech., 28 (1982), 25-36. doi: 10.1007/BF01230657.  Google Scholar

[5]

G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615. doi: 10.3934/dcds.2014.34.4589.  Google Scholar

[6]

S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. doi: 10.1007/BF02592182.  Google Scholar

[7]

R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175.  Google Scholar

[8]

R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM J. Math. Anal., 15 (1984), 857-876. doi: 10.1137/0515065.  Google Scholar

[9]

J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser Basel, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[10]

J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar

[11]

H. Poincaré, New Methods of Celestial Mechanics Vol. 1, American Institute of Physics, 1993.  Google Scholar

[12]

M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates, J. Phys. A: Math. Gen., 30 (1997), 5869-5876. doi: 10.1088/0305-4470/30/16/026.  Google Scholar

[13]

M. Shibayama, Non-integrability of the collinear three-body problem, Discrete Contin. Dyn. Syst., 30 (2011), 299-312. doi: 10.3934/dcds.2011.30.299.  Google Scholar

[14]

M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Nonlinearity, 22 (2009), 2377-2403. doi: 10.1088/0951-7715/22/10/004.  Google Scholar

[15]

H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems, Physica, 21 (1986), 163-170. doi: 10.1016/0167-2789(86)90087-4.  Google Scholar

[16]

H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9.  Google Scholar

[17]

M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians, RIMS Kokyuroku Bessatsu, B40 (2013), 177-189.  Google Scholar

[18]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I. Funktsional. Anal. i Prilozhen., 16 (1982), 30-41, 96.  Google Scholar

[19]

S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II. Funktsional. Anal. i Prilozhen., 17 (1983), 8-23.  Google Scholar

[1]

Naoyuki Ishimura, Shin'ya Matsui. On blowing-up solutions of the Blasius equation. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 985-992. doi: 10.3934/dcds.2003.9.985
[2]

Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929
[3]

Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021228
[4]

Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729
[5]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[6]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[7]

Shaodong Wang. Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary. Communications on Pure & Applied Analysis, 2018, 17 (1) : 209-230. doi: 10.3934/cpaa.2018013
[8]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589
[9]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020
[10]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
[11]

Jianqing Chen, Boling Guo. Sharp global existence and blowing up results for inhomogeneous Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 357-367. doi: 10.3934/dcdsb.2007.8.357
[12]

Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161
[13]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[14]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[15]

Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15
[16]

Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224
[17]

Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645
[18]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
[19]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[20]

Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences